Bertrand Russell suggested that the bigger a book, the more harm it can do; but this publication is a perfect counter example to Russell’s mischievous generality. Consisting of 718 pages and just three chapters, it forms a really attractive introduction to the mathematics and history of plane algebraic curves. And, although the authors state that it isn’t intended as an introduction to algebraic geometry, their book provides very good insight into that way of thinking.
The original German text first appeared in1978, followed by the English version in 1986. The translation, carried out by John Stillwell, is of such a high standard that one could be forgiven for thinking that English was the native tongue of the authors. The only mystery is why this book was out of circulation for such a long time — an anomaly now redressed by Birkhäuser, which has reprinted it in its “Modern Classics” series.
The introductory chapter is largely historical, but it simultaneously introduces the mathematical foundations for the development of the subsequent material. It is so elegantly written, and so attractively illustrated, that it almost forms a book in itself. Throughout the book, there are various facsimile extracts from original historical sources, such as Newton’s ‘Genesis of Curves by Shadows’, and there is early emphasis on the relevance of curves to optics, astronomy, kinematics and engineering (Watt’s curves and the rotary piston of the Wankel engine).
Having begun at high school level by reminding the reader of the basic ideas behind coordinate systems, this first chapter carefully escorts the reader into the world of complex projective geometry and provides a description of the projective plane as a manifold. As such, the first chapter (171 pages long) could form the basis of a course for maths majors at the intermediate undergraduate level.
For senior undergraduates, the essential mathematical techniques for the classification of quadrics and cubics in P2(C) are revealed over the 130 pages that make up the second chapter. The algebraic methods hinge upon the study of polynomials over the field of complex numbers. Emphasis is placed upon matters of factorisation, divisibility, zeroes, homogeneity and decomposition into irreducible components. With sufficient machinery in hand, geometrical considerations come to the fore with respect to the intersection of a line and a curve, diagnosis of singular points and further thoughts on intersection theory.
By the end of this second chapter, readers will appreciate the significance of Bézout’s theorem and will understand that the function theory of complex curves of low degree is almost synonymous with the theory of elliptic curves. Most aptly, a complete classification of quadrics and cubics emerges from this mathematical context.
In the case of irreducible cubics, one distinguishes between non-singular and two types of singular curves, depending on the type of singularity (e.g. double points or cusps). For curves of degree greater than 3, classification becomes increasingly difficult because of the complexity of the singularities. In fact, when the German version this book was published in 1978, the general problem of resolution of singularities was unsolved (it still is).
However, one technique for the resolution of singularities is to map plane algebraic curves to curves that have no singularity, but which may no longer reside in the projective plane. Indeed, such a curve may reside in an entirely different algebraic manifold — or perhaps no manifold whatever. The exploration of such questions involves the interplay between algebraic, analytic and topological methods for the study of these geometric properties. The last, and longest, chapter of this book is devoted to this theme.
Overall, this is a masterly expositional work in which the conversational style of narrative never leaves the reader in doubt about the direction of enquiry. The wealth of illustrations is a major attraction in itself; but the richness of this publication really resides in the fascinating range of mathematical ideas that support its main line of enquiry. As suggested, it can be read selectively at so many different levels up to the postgraduate stage. On the other hand, the extensive bibliography refers to nothing published later than 1976.
Peter Ruane spent most of his academic career in mathematics education (for primary and secondary school teachers).